To determine the factors of the polynomial [tex]\( x^3 + 4x^2 + 5x + 20 \)[/tex] by grouping, we can follow these steps:
1. Group the terms in pairs:
We start by dividing the polynomial into two groups. Group the first two terms together and the last two terms together:
[tex]\[
(x^3 + 4x^2) + (5x + 20)
\][/tex]
2. Factor out the common factor from each group:
- For the first group [tex]\( x^3 + 4x^2 \)[/tex], factor out [tex]\( x^2 \)[/tex]:
[tex]\[
x^2(x + 4)
\][/tex]
- For the second group [tex]\( 5x + 20 \)[/tex], factor out [tex]\( 5 \)[/tex]:
[tex]\[
5(x + 4)
\][/tex]
3. Combine the factored groups:
Notice that both terms now contain a common binomial factor [tex]\( (x + 4) \)[/tex]. We can factor [tex]\( (x + 4) \)[/tex] out from both terms:
[tex]\[
x^2(x + 4) + 5(x + 4)
\][/tex]
4. Rewrite the polynomial as a product of factors:
Now that we have factored out the common binomial, the polynomial [tex]\( x^3 + 4x^2 + 5x + 20 \)[/tex] can be written as:
[tex]\[
(x^2 + 5)(x + 4)
\][/tex]
Therefore, the correct way to show the factors of [tex]\( x^3 + 4x^2 + 5x + 20 \)[/tex] by grouping is:
[tex]\[
x^2(x + 4) + 5(x + 4)
\][/tex]
Thus, the correct option is:
[tex]\[
\boxed{x^2(x+4)+5(x+4)}
\][/tex]
